1. Sort beans and discard debris. Rinse beans, drain, and put into a 5- to 6-quart pan with 4 quarts water. Bring to a boil over high heat, cover, reduce heat, and simmer for 1 hour. Drain beans, discarding liquid; if making up to 1 day ahead, cover and chill. Put beans in a 4 1/2-quart or larger electric slow-cooker.
2. Meanwhile, pull and cut skin from duck breasts; reserve skin for cracklings (cover and chill up to 1 day). Place breasts in a bowl. Add 2 tablespoons salt and the sugar, and mix; cover and chill at least 30 minutes but no longer than 1 hour. Rinse breasts well under cool running water; if making up to 1 day ahead, cover and chill. Cut meat into 1/2-inch chunks.
3. In a 10- to 12-inch frying pan over medium-high heat, frequently stir bacon until browned and crisp, about 5 minutes. Discard all but 1 tablespoon bacon drippings. Add onions and garlic to bacon; stir often until onions begin to brown, 8 to 10 minutes. Add broth and wine; bring to a boil over high heat, scraping up browned bits from pan bottom. Pour onion-broth mixture into slow-cooker with beans. Add tomato, chopped thyme, pepper, and bay leaf; mix. Place duck pieces on the beans.
4. Cover and cook until beans and duck are very tender to bite, 5 to 6 hours on low, 4 to 5 hours on high. If there is more liquid than desired, uncover slow-cooker, turn heat to high, and simmer to concentrate the mixture, up to 30 minutes.
5. Stir sausage slices into cassoulet and cook until hot, 5 to 10 minutes. Ladle cassoulet into bowls, sprinkle equally with duck cracklings, and garnish with thyme sprigs.
Nutritional analysis per serving without duck cracklings.
Duck cracklings. Chop reserved duck skin (see preceding) into 1/4-inch pieces. In a 10- to 12-inch frying pan over medium-high heat, frequently stir skin until crisp and golden, 5 to 8 minutes. With a slotted spoon, transfer to towels to drain. Sprinkle with salt. When cool, wrap airtight; chill up to 1 day. Makes 1/3 to 1/2 cup.